How to Calculate Tension in Physics

Step-by-Step Guide

1. 1

   Step 1: Define the forces on either end of the strand.

   The tension in a given strand of string or rope is a result of the forces pulling on the rope from either end.
   
   As a reminder, force = mass × acceleration.
   
   Assuming the rope is stretched tightly, any change in acceleration or mass in objects the rope is supporting will cause a change in tension in the rope.
   
   Don't forget the constant acceleration due to gravity
   - even if a system is at rest, its components are subject to this force.
   
   We can think of a tension in a given rope as T = (m × g) + (m × a), where "g" is the acceleration due to gravity of any objects the rope is supporting and "a" is any other acceleration on any objects the rope is supporting.
   
   For the purposes of most physics problems, we assume ideal strings
   - in other words, that our rope, cable, etc. is thin, massless, and can't be stretched or broken.
   
   As an example, let's consider a system where a weight hangs from a wooden beam via a single rope (see picture).
   
   Neither the weight nor the rope are moving
   - the entire system is at rest.
   
   Because of this, we know that, for the weight to be held in equilibrium, the tension force must equal the force of gravity on the weight.
   
   In other words, Tension (Ft) = Force of gravity (Fg) = m × g.
   
   Assuming a 10 kg weight, then, the tension force is 10 kg ×
   9.8 m/s2 = 98 Newtons.
2. 2

   Step 2: Account for acceleration.

   Gravity isn't the only force that can affect the tension in a rope
   - so can any force related to acceleration of an object the rope is attached to.
   
   If, for instance, a suspended object is being accelerated by a force on the rope or cable, the acceleration force (mass × acceleration) is added to the tension caused by the weight of the object.
   
   Let's say that, in our example of the 10 kg weight suspended by a rope, that, instead of being fixed to a wooden beam, the rope is actually being used to pull the weight upwards at an acceleration of 1 m/s2.
   
   In this case, we must account for the acceleration on the weight as well as the force of gravity by solving as follows:
   Ft = Fg + m × a Ft = 98 + 10 kg × 1 m/s2 Ft = 108 Newtons. , An object being rotated around a central point via a rope (like a pendulum) exerts strain on the rope caused by centripetal force.
   
   Centripetal force is the added tension force the rope exerts by "pulling" inward to keep an object moving in its arc and not in a straight line.
   
   The faster the object is moving, the greater the centripetal force.
   
   Centripetal force (Fc) is equal to m × v2/r where "m" is mass, "v" is velocity, and "r" is the radius of the circle that contains the arc of the object's motion.
   
   Since the direction and magnitude of centripetal force changes as the object on the rope moves and changes speeds, so does the total tension in the rope, which always pulls parallel to the rope towards the central point.
   
   Remember also that the force of gravity is constantly acting on the object in a downward direction.
   
   So, if an object is being spun or swung vertically, total tension is greatest at the bottom of the arc (for a pendulum, this is called the equilibrium point) when the object is moving fastest and least at the top of the arc when it is moving slowest.
   
   Let's say in our example problem that our object is no longer accelerating upwards but instead is swinging like a pendulum.
   
   We'll say that our rope is
   1.5 meters (4.9 ft) long and that our weight is moving at 2 m/s when it passes through the bottom of its swing.
   
   If we want to calculate tension at the bottom of the arc when it's highest, we would first recognize that the tension due to gravity at this point is the same as when the weight was held motionless
   - 98 Newtons.To find the additional centripetal force, we would solve as follows:
   Fc = m × v2/r Fc = 10 × 22/1.5 Fc =10 ×
   2.67 =
   26.7 Newtons.
   
   So, our the total tension would be 98 +
   26.7 =
   124.7 Newtons. , As noted above, both the direction and magnitude of centripetal force change as an object swings.
   
   However, though the force of gravity remains constant, the tension resulting from gravity also changes.
   
   When a swinging object isn't at the bottom of its arc (its equilibrium point), gravity is pulling directly downward, but tension is pulling up at an angle.
   
   Because of this, tension only has to counteract part of the force due to gravity, rather than its entirety.
   
   Breaking gravitational force up into two vectors can help you visualize this concept.
   
   At any given point in the arc of a vertically swinging object, the rope forms an angle "θ" with the line through the equilibrium point and the central point of rotation.
   
   As the pendulum swings, gravitational force (m × g) can be broken up into two vectors
   - mgsin(θ) acting tangent to the arc in the direction of the equilibrium point and mgcos(θ) acting parallel to the tension force in the opposite direction.
   
   Tension only has to counter mgcos(θ)
   - the force pulling against it
   - not the entire gravitational force (except at the equilibrium point, when these are equal).
   
   Let's say that when our pendulum forms an angle of 15 degrees with the vertical, it's moving
   1.5 m/s.
   
   We would find tension by solving as follows:
   Tension due to gravity (Tg) = 98cos(15) = 98(0.96) =
   94.08 Newtons Centripetal force (Fc) = 10 ×
   1.52/1.5 = 10 ×
   1.5 = 15 Newtons Total tension = Tg + Fc =
   94.08 + 15 =
   109.08 Newtons. , Any object being pulled by a rope that experiences a "drag" force from friction against another object (or fluid) transfers this force to the tension in the rope.
   
   Force from friction between two objects is calculated as it would be in any other situation
   - via the following equation:
   Force due to friction (usually written Fr) = (mu)N, where mu is the friction coefficient between the two objects and N is the normal force between the two objects, or the force with which they are pressing into each other.
   
   Note that static friction
   - the friction that results when trying to put a stationary object into motion
   - is different than kinetic friction
   - the friction that results when trying to keep a moving object in motion.
   
   Let's say that our 10 kg weight is no longer being swung but is now being dragged horizontally along the ground by our rope.
   
   Let's say that the ground has a kinetic friction coefficient of
   0.5 and that our weight is moving at a constant velocity but that we want to accelerate it at 1 m/s2.
   
   This new problem presents two important changes
   - first, we no longer have to calculate tension due to gravity because our rope isn't supporting the weight against its force.
   
   Second, we have to account for tension caused by friction, as well as that caused by accelerating the weight's mass.
   
   We would solve as follows:
   Normal force (N) = 10 kg ×
   9.8 (acceleration from gravity) = 98 N Force from kinetic friction (Fr) =
   0.5 × 98 N = 49 Newtons Force from acceleration (Fa) = 10 kg × 1 m/s2 = 10 Newtons Total tension = Fr + Fa = 49 + 10 = 59 Newtons.
3. 3

   Step 3: Account for rotational acceleration.

4. 4

   Step 4: Understand that tension due to gravity changes throughout a swinging object's arc.

5. 5

   Step 5: Account for friction.

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